HOMEWORK 1
Shruti Bhargava – 13671
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OBJECTIVE – To analyse the given data representing height measurements for dropping a ball down for a fixed amount of time by demonstrating the possibility of fitting polynomial curves for the same.
POLYNOMIAL COEFFICIENTS obtained as BEST-FIT on 'train_small' DATA
|
COEFFICIENT OF |
Degree=0 |
Degree=1 |
Degree=2 |
Degree=3 |
Degree=5 |
Degree=9 |
|
x0 |
156.450067204000 |
-82.8186759849698 |
27.8869657019996 |
-85.1430926667695 |
-6420.09625284803 |
2079030.46157192 |
|
x1 |
|
50.3723669871515 |
-0.930247453151366 |
79.6796048115152 |
7804.79521687200 |
-4377092.01233501 |
|
x2 |
|
|
5.40027520424241 |
-12.5410039019114 |
-3647.89865004070 |
4028322.83250255 |
|
x3 |
|
|
|
1.25903713025641 |
827.588111266824 |
-2128283.16321080 |
|
x4 |
|
|
|
|
-90.9093007601595 |
711845.770445245 |
|
x5 |
|
|
|
|
3.88200060455207 |
-156416.601522000 |
|
x6 |
|
|
|
|
|
22595.0216489479 |
|
x7 |
|
|
|
|
|
-2070.43677692882 |
|
|
|
|
|
|
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109.268828731187 |
|
|
|
|
|
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-2.53198416266513 |
POLYNOMIAL COEFFICIENTS obtained as BEST-FIT on 'train_big' DATA
|
COEFFICIENT OF |
Degree=0 |
Degree=1 |
Degree=2 |
Degree=3 |
Degree=5 |
Degree=9 |
|
x0 |
256.383861985551 |
-166.283574829398 |
-19.4993885696681 |
27.5620995195070 |
353.084080659908 |
2168.67496261319 |
|
x1 |
|
67.6267898903919 |
14.0559189926803 |
-12.8235869232669 |
-321.411225178492 |
-3575.61215516570 |
|
x2 |
|
|
4.28566967181692 |
8.92958994901994 |
119.167071304045 |
2619.87240619193 |
|
x3 |
|
|
|
-0.247675748117494 |
-18.9145331882118 |
-1098.08972536985 |
|
x4 |
|
|
|
|
1.50850817329284 |
289.251573863219 |
|
x5 |
|
|
|
|
-0.0468359692419199 |
-49.1433442618434 |
|
x6 |
|
|
|
|
|
5.35551049099945 |
|
x7 |
|
|
|
|
|
-0.359756172388189 |
|
x8 |
|
|
|
|
|
0.0134830214286634 |
|
x9 |
|
|
|
|
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-0.000214195736217227 |
The above polynomials have been obtained by using the least squares method.
OBSERVATION- The coefficients switch and increase in magnitude with the increasing degree.
Plot comparing Accuracy on TEST, VALIDATION and small TRAINING data
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"
name="Image2" align="right" width="71%"
border="0">
Illustration
1: Plot showing RMS Error vs Degree of polynomial
OBSERVATIONS:-
From the graph we infer that for the training data, the error is decreasing as the degree of the fitting polynomial increases.
For degree=9, the error is 0. This is because the training set included exactly 9 points and hence the polynomial passes through all of these 9 points.
For the test set (which comprises of 35 data points), initially the error reduces with increasing the degree of polynomial, with a minima at 2. However on making the graph further flexible, ie increasing the degree, the error obtained from the polynomial increases drastically. This is due to the fact that the higher degree polynomials are very oscillatory. And hence diverge at a very rapid rate.
For the validation data, the error reduces and a minimum is again obtained at 2, the error here being smaller than that obtained on the training set itself. On increasing the degree, the error soon reaches very high values.
Thereby we observe that the accuracy in training data increases in case of higher degree polynomials but it has a reverse effect on validation and test data. Due to OVERFITTING(a result of the oscillatory nature of polynomials), the errors are much higher in the higher polynomials
RESULT:-
The optimal degree to fit a polynomial on a given data set varies with the size of the data. The most important factor being that the number of data sets should be comparatively larger than the degree, in order to avoid overfitting and get useful results.
In this case the optimal degree is 2.
The error obtained on the test set in our optimal model is 24.56483.
BIG vs SMALL TRAINING SET
OBSERVATIONS:-
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"
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Illustration
2: Comparing accuracy (using test data) of polynomials generated
from small and big training sets
The accuracy increased (error reduced) for each polynomial set.
The increase in accuracy for the higher degrees is very drastic.
The minimum error is still observed in polynomial of degree 2.
INFERENCE:-
Larger the size of training set (or database of prior information) , better a higher degree polynomial fits into it!
This training set also suggests that a 2 degree polynomial (ie quadratic) works most accurately.
In this case, since the data available is larger, we can use cross validation. ie. We can take different sets as the validation set in different instances. This would enable us to have a better check on the accuracy.
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ANALYSIS
In the analysis, we conclude the following major points :-
Illustration
3: Error analysis of training set on the polynomials
Ovefitting- caused due to oscillatory nature of polynomials – often results in large errors when tested on data other than the training set.
This error can be somewhat reduced by using a large training set and ensuring that the number of elements in the set is much more than the degree of the polynomial.
Larger the training set, easier it is to fit in complex(higher degree) and more flexible polynomials.
Validation set is a part of the training data which one keeps apart in order to check the corrrectness of the functions derived and optimize the model complexity.
while
Test set is an unknown data on which our inferences derived are later checked and analysed.
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Quadratic polynomial obtained from the bigger training set (in the table above), since it has the least error on the test set. |
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VERIFICATION |
Since the data available to us was for a bouncing ball, hence there actu near ally exists a near quadratic relation between it's height and time- laws of motion give the relation- s = u*t+a*t*t/2 .